Question: $ 0.\overline{83} \div 3.\overline{4} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 83.8383...\\ x &= 0.8383...\end{align*} $ $\begin{align*} 99x &= 83 \\ x &= \dfrac{83}{99}\end{align*} $ $\begin{align*} 10y &= 34.4444...\\ y &= 3.4444...\end{align*} $ $\begin{align*} 9y &= 31 \\ y &= \dfrac{31}{9}\end{align*} $ So, the problem becomes: $ \dfrac{83}{99} \div \dfrac{31}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{83}{99} \times \dfrac{9}{31} = {?} $ $ \phantom{\dfrac{83}{99} \times \dfrac{31}{9}} = \dfrac{83 \times 9}{99 \times 31} $ $ \phantom{\dfrac{83}{99} \times \dfrac{31}{9}} = \dfrac{83 \times \cancel{9}} {\cancel{99}11 \times 31} $ $ \phantom{\dfrac{83}{99} \times \dfrac{31}{9}} = \dfrac{83}{341} $